Is there a preferred solution among the solutions to Schrödinger's eq?

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Discussion Overview

The discussion centers on the nature of solutions to Schrödinger's equation, particularly the implications of its deterministic evolution and the existence of multiple solutions. Participants explore the relationship between initial conditions and the uniqueness of solutions in the context of quantum mechanics.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants express confusion regarding the deterministic nature of Schrödinger's equation, questioning how a preferred solution can exist when the equation has many solutions.
  • One participant asserts that the Schrödinger equation is deterministic, stating that given initial conditions, the solution at later times is uniquely determined.
  • Another participant suggests that Schrödinger's equation has an infinite number of solutions, specifically mentioning linear combinations of plane waves.
  • A different participant emphasizes the necessity of initial conditions for solving the Schrödinger equation, noting that without them, the problem is not uniquely defined.
  • There is acknowledgment that once initial conditions are provided, the solutions become unique rather than infinite.

Areas of Agreement / Disagreement

Participants do not reach consensus on whether there is a preferred solution to Schrödinger's equation. While some argue for the uniqueness of solutions given initial conditions, others maintain that the equation inherently allows for multiple solutions.

Contextual Notes

The discussion highlights the dependence on initial conditions for defining solutions to the Schrödinger equation and the implications this has for the interpretation of quantum mechanics. There are unresolved aspects regarding the nature of solutions and their implications for determinism.

nomadreid
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I am a little confused when it is stated that Schrödinger's equation represents a deterministic evolution of the wave function of a particle. I would be OK with the idea that time evolution went from one state to another state in a deterministic way (even though each state, with respect to its eigenvalues, is not deterministic) if there were only one solution to the equation. However, it has many solutions, so why would there be a preferred solution?
 
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nomadreid said:
I am a little confused when it is stated that Schrödinger's equation represents a deterministic evolution of the wave function of a particle. I would be OK with the idea that time evolution went from one state to another state in a deterministic way (even though each state, with respect to its eigenvalues, is not deterministic) if there were only one solution to the equation. However, it has many solutions, so why would there be a preferred solution?
I don't understand your remark, because leaving aside the issue of what happens during a measurement, the Schrödinger equation itself is deterministic. The time-dependent Schrödinger equation is a first order differential equation in time, and if you're given initial values everywhere at t = 0, the solution at times t > 0 is uniquely determined. There's one and only one solution.
 
Thanks for the reply, Bill_K. Your answer then takes me from the frying pan into the fire. I thought --and this is probably wrong -- Schrödinger's equation is a differential equation which has an infinite number of solutions: any linear combination of plane waves.
 
nomadreid, Just like any differential equation, you have to be given initial conditions before you can write down the solution. You don't know how high a projectile will go until you are told its initial position and velocity. Likewise to solve the Schrödinger equation, you have to be given its initial values. And since it's a partial differential equation, you have to be given the value of ψ(x, 0) everywhere, at the initial time t = 0. Without being told this information, the problem is not uniquely defined. And it's not the fault of the Schrödinger equation especially, that's just the way PDEs work. But given these initial conditions, there is no longer an infinity of different possible solutions - only one.
 
Last edited:
Bill_K. Thanks very much. Initial conditions, right. Makes sense.
 

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